Approximation of non-linear kernels using random feature mapping has been successfully employed in large-scale data analysis applications, accelerating the training of kernel machines. While previous random feature mappings run in O(ndD) time for n training samples in d-dimensional space and D random feature maps, we propose a novel randomized tensor product technique, called Tensor Sketching, for approximating any polynomial kernel in O(n(d + D log D)) time. Also, we introduce both absolute and relative error bounds for our approximation to guarantee the reliability of our estimation algorithm. Empirically, Tensor Sketching achieves higher accuracy and often runs orders of magnitude faster than the state-of-the-art approach for large-scale real-world datasets.

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Estimating set similarity is a central problem in many computer applications. In this paper we introduce the Odd Sketch, a compact binary sketch for estimating the Jaccard similarity of two sets. The exclusive-or of two sketches equals the sketch of the symmetric difference of the two sets. This means that Odd Sketches provide a highly spaceefficient estimator for sets of high similarity, which is relevant in applications such as web duplicate detection, collaborative filtering, and association rule learning. The method extends to weighted Jaccard similarity, relevant e.g. for TF-IDF vector comparison.We present a theoretical analysis of the quality of estimation to guarantee the reliability of Odd Sketch-based estimators. Our experiments confirm this efficiency, and demonstrate the efficiency of Odd Sketches in comparison with b-bit minwise hashing schemes on association rule learning and web duplicate detection tasks.

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